A seminar series.
Matthew Krauel
(Sacramento State)
In Search of New Algebraic Structures, Part I
Loosely speaking, representation theory is the study of algebraic structures by representing their elements as matrices and its product as the product of matrices. This essentially reduces many problems to linear algebra. It is an overarching goal in mathematical theories to classify representations for a given algebraic structure. Often, knowledge of a sub-structure can be used in this process, and in particular to create new representations by utilizing representations of the sub-structure. However, in some areas of mathematics and physics, such as 2-dimensional conformal field theory, this process is opaque.
In this talk, I will introduce the concepts of representations in the theory of groups and describe some of the fundamental machinery in this area. I will then proceed to explain how this theory also works (or doesn't) with other algebraic structures. The goal will be to build towards explaining an open problem that persists among a number of important algebraic structures, and in particular Lie algebras, where an answer would have ramifications in conformal field theory.
Matthew Krauel
(Sacramento State)
In Search of New Algebraic Structures, Part II
In this talk, I will continue discussing an open problem concerning the representation theory of Lie algebras. In particular, I will explain how new algebraic structures can be found and time permitting, I will provide some rational on what governs such structures.
Gabriel Martins
(MSRI)
Magnetic Confinement from a Dynamical Perspective
We study how to use a magnetic field to trap charged particles inside a bounded region in space with smooth boundary; this provides a simple model for the interior of fusion reactor devices like Tokamaks and Stellarators. We start by discussing the concept of an electromagnetic field and recalling Maxwell's equations. We then discuss the Lorentz force and the Hamiltonian structure of this problem. Finally we describe a few examples where one can prove that particles become confined to the interior of the region for all time.